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507
The Intrinsic Normal Cone
 Invent. Math
, 1997
"... We suggest a construction of virtual fundamental classes of certain types of moduli spaces. Contents 0 ..."
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Cited by 238 (7 self)
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We suggest a construction of virtual fundamental classes of certain types of moduli spaces. Contents 0
Higher topos theory
, 2006
"... Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain com ..."
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Cited by 78 (1 self)
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Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain complex of Gvalued singular cochains on X. An alternative is to regard H n (•, G) as a representable functor on the homotopy category
Twisted Ktheory of differentiable stacks
 ANN. SCI. ÉCOLE NORM. SUP
, 2004
"... In this paper, we develop twisted Ktheory for stacks, where the twisted class is given by an S 1gerbe over the stack. General properties, including the Mayer–Vietoris property, Bott periodicity, and the product structure K i α ⊗K j β → Ki+j α+β are derived. Our approach provides a uniform framew ..."
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Cited by 55 (12 self)
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In this paper, we develop twisted Ktheory for stacks, where the twisted class is given by an S 1gerbe over the stack. General properties, including the Mayer–Vietoris property, Bott periodicity, and the product structure K i α ⊗K j β → Ki+j α+β are derived. Our approach provides a uniform framework for studying various twisted Ktheories including the usual twisted Ktheory of topological spaces, twisted equivariant Ktheory, and the twisted Ktheory of orbifolds. We also present a Fredholm picture, and discuss the conditions under which twisted Kgroups can be expressed by socalled “twisted vector bundles”. Our approach is to work on presentations of stacks, namely groupoids, and relies heavily on the machinery of Ktheory (KKtheory) of C ∗algebras.
Local projective model structures on simplicial presheaves
 Ktheory
"... Abstract. We give a model structure on the category of simplicial presheaves on some essentially small Grothendieck site T. When T is the Nisnevich site it specializes to a proper simplicial model category with the same weak equivalences as in [MV], but with fewer cofibrations and consequently more ..."
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Cited by 40 (0 self)
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Abstract. We give a model structure on the category of simplicial presheaves on some essentially small Grothendieck site T. When T is the Nisnevich site it specializes to a proper simplicial model category with the same weak equivalences as in [MV], but with fewer cofibrations and consequently more fibrations. This allows a simpler proof of the comparison theorem of [V2], one which makes no use of ∆closed classes. The purpose of this note is to introduce different model structures on the categories of simplicial presheaves and simplicial sheaves on some essentially small Grothendieck site T and to give some applications of these simplified model categories. In particular, we prove that the stable homotopy categories SH((Sm/k)Nis, A 1) and SH((Sch/k)cdh, A 1) are equivalent. This result was first proven by Voevodsky in [V2] and our proof uses many of his techniques, but it does not use his theory of ∆closed classes developed in [V3]. 1. The local projective model structure on presheaves We first recall some of the other wellknown model structures on simplicial presheaves. Definition 1.1. A map f: X → Y of simplicial presheaves (or sheaves) is a local weak equivalence if f ∗ : π0(X) → π0(Y) induces an isomorphism of associated sheaves and, for all U ∈ T, f ∗ : πn(X, x) → πn(Y, f(x)) induces an isomorphism of associated sheaves on T/U for any choice of basepoint x ∈ X(U). The map f is a sectionwise weak equivalence (respectively sectionwise fibration) if for all U ∈ T, the map f(U) : X(U) → Y (U) is a weak equivalence (respectively Kan fibration) of simplicial sets. Heller [He] discovered a model structure on simplicial presheaves whose weak equivalences are the sectionwise weak equivalences. We will refer to his model structure as the injective model structure. Date: January 11, 2001. I would like to thank Dan Isaksen for his many helpful suggestions, and I thank my adviser Peter May for his encouragement and careful reading of many drafts. I am also grateful to Vladimir Voevodsky for noticing an error in an earlier version and for his work that inspired this note. 1 2 BENJAMIN BLANDER
Hypercovers and simplicial presheaves
 MATH. PROC. CAMBRIDGE PHILOS. SOC
, 2004
"... We use hypercovers to study the homotopy theory of simplicial presheaves. The main result says that model structures for simplicial presheaves involving local weak equivalences can be constructed by localizing at the hypercovers. One consequence is that the fibrant objects can be explicitly describe ..."
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Cited by 40 (6 self)
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We use hypercovers to study the homotopy theory of simplicial presheaves. The main result says that model structures for simplicial presheaves involving local weak equivalences can be constructed by localizing at the hypercovers. One consequence is that the fibrant objects can be explicitly described in terms of a hypercover descent condition. These ideas are central to constructing realization functors on the homotopy theory of schemes [DI1, Is]. We give a few other applications for this new description of the homotopy theory of simplicial presheaves.
Smashing Subcategories And The Telescope Conjecture  An Algebraic Approach
 Invent. Math
, 1998
"... . We prove a modified version of Ravenel's telescope conjecture. It is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra. This result is based on a new characterization of smashing subcategories, which leads in addition to ..."
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. We prove a modified version of Ravenel's telescope conjecture. It is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra. This result is based on a new characterization of smashing subcategories, which leads in addition to a classification of these subcategories in terms of the category of finite spectra. The approach presented here is purely algebraic; it is based on an analysis of pureinjective objects in a compactly generated triangulated category, and covers therefore also situations arising in algebraic geometry and representation theory. Introduction Smashing subcategories naturally arise in the stable homotopy category S from localization functors l : S ! S which induce for every spectrum X a natural isomorphism l(X) ' X l(S) between the localization of X and the smash product of X with the localization of the sphere spectrum S. In fact, a localization functor has this property if and only if it preserv...